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Youness Lamzouri

Publications and source records attributed to Youness Lamzouri.

At least 37 records · Page 2Linked to original sources
arXiv2020

Small values of $| L^\prime/L(1,χ) |$

In this paper, we investigate the quantity $m_q:=\min_{χ\ne χ_0} | L^\prime/L(1,χ)|$, as $q\to \infty$ over the primes, where $L(s,χ)$ is the Dirichlet $L$-function attached to a non trivial Dirichlet character modulo $q$. Our main result shows that $m_q \ll \log\log q/\sqrt{\log q}$. We also compute $m_q$ for every odd prime $q$ up to $10^7$. As a consequence we numerically verified that for every odd prime $q$, $3 \le q \le 10^7$, we have $c_1/q< m_q<5/\sqrt{q}$, with $c_1=21/200$. In particular, this shows that $L^\prime(1,χ) \ne 0$ for every non trivial Dirichlet character $χ$ mod $q$ where $3\leq q\leq 10^7$ is prime, answering a question of Gun, Murty and Rath in this range. We also provide some statistics and scatter plots regarding the $m_q$-values, see Section 6. The programs used and the computational results described here are available at the following web address: \url{http://www.math.unipd.it/~languasc/smallvalues.html}.

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arXiv2019

Zeros of the Epstein zeta function to the right of the critical line

Let $E(s, Q)$ be the Epstein zeta function attached to a positive definite quadratic form of discriminant $D<0$, such that $h(D)\geq 2$, where $h(D)$ is the class number of the imaginary quadratic field $\mathbb{Q}(\sqrt{D})$. We denote by $N_E(σ_1, σ_2, T)$ the number of zeros of $E(s, Q)$ in the rectangle $σ_1 <\text{Re}(s)\leq σ_2$ and $T\leq \text{Im}(s)\leq 2T$, where $1/2<σ_1<σ_2<1$ are fixed real numbers. In this paper, we improve the asymptotic formula of Gonek and Lee for $N_E(σ_1, σ_2, T)$, obtaining a saving of a power of $\log T$ in the error term.

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arXiv2018

On the distribution of the maximum of cubic exponential sums

In this paper, we investigate the distribution of the maximum of partial sums of certain cubic exponential sums, commonly known as "Birch sums". Our main theorem gives upper and lower bounds (of nearly the same order of magnitude) for the distribution of large values of this maximum, that hold in a wide uniform range. This improves a recent result of Kowalski and Sawin. The proofs use a blend of probabilistic methods, harmonic analysis techniques, and deep tools from algebraic geometry. The results can also be generalized to other types of $\ell$-adic trace functions. In particular, the lower bound of our result also holds for partial sums of Kloosterman sums. As an application, we show that there exist $x\in [1, p]$ and $a\in \mathbb{F}_p^{\times}$ such that $|\sum_{n\leq x} \exp(2πi (n^3+an)/p)|\ge (2/π+o(1)) \sqrt{p}\log\log p$. The uniformity of our results suggests that this bound is optimal, up to the value of the constant.

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arXiv2017

Extreme biases in prime number races with many contestants

We continue to investigate the race between prime numbers in many residue classes modulo $q$, assuming the standard conjectures GRH and LI. We show that provided $n/\log q \rightarrow \infty$ as $q \rightarrow \infty$, we can find $n$ competitor classes modulo $q$ so that the corresponding $n$-way prime number race is extremely biased. This improves on the previous range $n \geq φ(q)^ε$, and (together with an existing result of Harper and Lamzouri) establishes that the transition from all $n$-way races being asymptotically unbiased, to biased races existing, occurs when $n = \log^{1+o(1)}q$. The proofs involve finding biases in certain auxiliary races that are easier to analyse than a full $n$-way race. An important ingredient is a quantitative, moderate deviation, multi-dimensional Gaussian approximation theorem, which we prove using a Lindeberg type method.

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arXiv2017

On the average of the number of imaginary quadratic fields with a given class number

Let $\mathcal{F}(h)$ be the number of imaginary quadratic fields with class number $h$. In this note, we improve the error term in Soundararajan's asymptotic formula for the average of $\mathcal{F}(h)$. Our argument leads to a similar refinement of the asymptotic for the average of $\mathcal{F}(h)$ over odd $h$, which was recently obtained by Holmin, Jones, Kurlberg, McLeman and Petersen.

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arXiv2017

Large sums of Hecke eigenvalues of holomorphic cusp forms

Let $f$ be a Hecke cusp form of weight $k$ for the full modular group, and let $\{λ_f(n)\}_{n\geq 1}$ be the sequence of its normalized Fourier coefficients. Motivated by the problem of the first sign change of $λ_f(n)$, we investigate the range of $x$ (in terms of $k$) for which there are cancellations in the sum $S_f(x)=\sum_{n\leq x} λ_f(n)$. We first show that $S_f(x)=o(x\log x)$ implies that $λ_f(n)<0$ for some $n\leq x$. We also prove that $S_f(x)=o(x\log x)$ in the range $\log x/\log\log k\to \infty$ assuming the Riemann hypothesis for $L(s, f)$, and furthermore that this range is best possible unconditionally. More precisely, we establish the existence of many Hecke cusp forms $f$ of large weight $k$, for which $S_f(x)\gg_A x\log x$, when $x=(\log k)^A.$ Our results are $GL_2$ analogues of work of Granville and Soundararajan for character sums, and could also be generalized to other families of automorphic forms.

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arXiv2017

Large moments and extreme values of class numbers of indefinite binary quadratic forms

Let $h(d)$ be the class number of indefinite binary quadratic forms of discriminant $d$, and let $\varepsilon_d$ be the corresponding fundamental unit. In this paper, we obtain an asymptotic formula for the $k$-th moment of $h(d)$ over positive discriminants $d$ with $\varepsilon_d\leq x$, uniformly for real numbers $k$ in the range $0<k\leq (\log x)^{1-o(1)}$. This improves upon the work of Raulf, who obtained such an asymptotic for a fixed positive integer $k$. We also investigate the distribution of large values of $h(d)$ when the $d$'s are ordered according to the size of their fundamental units $\varepsilon_d$. In particular, we show that the tail of this distribution has the same shape as that of class numbers of imaginary quadratic fields ordered by the size of their discriminants. As an application of these results, we prove that there are many positive discriminants $d$ with class number $h(d)\geq (e^γ/3+o(1))\cdot\varepsilon_d (\log\log \varepsilon_d)/\log \varepsilon_d$, a bound that we believe is best possible. We also obtain an upper bound for $h(d)$ that is twice as large, assuming the generalized Riemann hypothesis.

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arXiv2017

Large odd order character sums and improvements of the Pólya-Vinogradov inequality

For a primitive Dirichlet character $χ$ modulo $q$, we define $M(χ)=\max_{t } |\sum_{n \leq t} χ(n)|$. In this paper, we study this quantity for characters of a fixed odd order $g\geq 3$. Our main result provides a further improvement of the classical Pólya-Vinogradov inequality in this case. More specifically, we show that for any such character $χ$ we have $$M(χ)\ll_{\varepsilon} \sqrt{q}(\log q)^{1-δ_g}(\log\log q)^{-1/4+\varepsilon},$$ where $δ_g := 1-\frac{g}π\sin(π/g)$. This improves upon the works of Granville and Soundararajan and of Goldmakher. Furthermore, assuming the Generalized Riemann hypothesis (GRH) we prove that $$ M(χ) \ll \sqrt{q} \left(\log_2 q\right)^{1-δ_g} \left(\log_3 q\right)^{-\frac{1}{4}}\left(\log_4 q\right)^{O(1)}, $$ where $\log_j$ is the $j$-th iterated logarithm. We also show unconditionally that this bound is best possible (up to a power of $\log_4 q$). One of the key ingredients in the proof of the upper bounds is a new Halász-type inequality for logarithmic mean values of completely multiplicative functions, which might be of independent interest.

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arXiv2016

Conditional bounds for the least quadratic non-residue and related problems

This paper studies explicit and theoretical bounds for several interesting quantities in number theory, conditionally on the Generalized Riemann Hypothesis. Specifically, we improve the existing explicit bounds for the least quadratic non-residue and the least prime in an arithmetic progression. We also refine the classical conditional bounds of Littlewood for $L$-functions at $s=1$. In particular, we derive explicit upper and lower bounds for $L(1,χ)$ and $ζ(1+it)$, and deduce explicit bounds for the class number of imaginary quadratic fields. Finally, we improve the best known theoretical bounds for the least quadratic non-residue, and more generally, the least $k$-th power non-residue.

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arXiv2016

An effective universality theorem for the Riemann zeta-function

Let $0 0$, once $T$ is large enough. This was refined by Bagchi who showed that the measure of such $t \in [0,T]$ is $(c(\varepsilon) + o(1)) T$, for all but at most countably many $\varepsilon > 0$. Using a completely different approach, we obtain the first effective version of Voronin's Theorem, by showing that in the rate of convergence one can save a small power of the logarithm of $T$. Our method is flexible, and can be generalized to other $L$-functions in the $t$-aspect, as well as to families of $L$-functions in the conductor aspect.

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arXiv2016

The distribution of class numbers in a special family of real quadratic fields

We investigate the distribution of class numbers in the family of real quadratic fields $\mathbb{Q}(\sqrt{d})$ corresponding to fundamental discriminants of the form $d=4m^2+1$, which we refer to as Chowla's family. Our results show a strong similarity between the distribution of class numbers in this family and that of class numbers of imaginary quadratic fields. As an application of our results, we prove that the average order of the number of quadratic fields in Chowla's family with class number $h$ is $(\log h)/2G$, where $G$ is Catalan's constant. With minor modifications, one can obtain similar results for Yokoi's family of real quadratic fields $\mathbb{Q}(\sqrt{d})$, which correspond to fundamental discriminants of the form $d=m^2+4$.

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arXiv2015

Orderings of weakly correlated random variables, and prime number races with many contestants

We investigate the race between prime numbers in many residue classes modulo $q$, assuming the standard conjectures GRH and LI. Among our results we exhibit, for the first time, prime races modulo $q$ with $n$ competitor classes where the biases do not dissolve when $n, q\to \infty$. We also study the leaders in the prime number race, obtaining asymptotic formulae for logarithmic densities when the number of competitors can be as large as a power of $q$, whereas previous methods could only allow a power of $\log q$. The proofs use harmonic analysis related to the Hardy--Littlewood circle method to control the average size of correlations in prime number races. They also use various probabilistic tools, including an exchangeable pairs version of Stein's method, normal comparison tools, and conditioning arguments. In the process we derive some general results about orderings of weakly correlated random variables, which may be of independent interest.

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arXiv2015

Large values of $L(1,χ)$ for $k$-th order characters $χ$ and applications to character sums

For any given integer $k\geq 2$ we prove the existence of infinitely many $q$ and characters $ χ\pmod q$ of order $k$, such that $|L(1,χ)|\geq (e^γ+o(1))\log\log q$. We believe this bound to be best possible. When the order $k$ is even, we obtain similar results for $L(1,χ)$ and $L(1,χξ)$ where $χ$ is restricted to even (or odd) characters of order $k$, and $ξ$ is a fixed quadratic character. As an application of these results, we exhibit large even order character sums, which are likely to be optimal.

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arXiv2015

A bias in Mertens' product formula

Rosser and Schoenfeld remarked that the product $\prod_{p\leq x}(1-1/p)^{-1}$ exceeds $e^γ \log x$ for all $2\leq x\leq 10^8$, and raised the question whether the difference changes sign infinitely often. This was confirmed in a recent paper of Diamond and Pintz. In this paper, we show (under certain hypotheses) that there is a strong bias in the race between the product $\prod_{p\leq x}(1-1/p)^{-1}$ and $e^γ\log x$ which explains the computations of Rosser and Schoenfeld.

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arXiv2015

Extreme values of class numbers of real quadratic fields

We improve a result of H. L. Montgomery and J. P. Weinberger by establishing the existence of infinitely many fundamental discriminants $d>0$ for which the class number of the real quadratic field $\mathbb{Q}(\sqrt{d})$ exeeds $(2e^γ+o(1)) \sqrt{d}(\log\log d)/\log d$. We believe this bound to be best possible. We also obtain upper and lower bounds of nearly the same order of magnitude, for the number of real quadratic fields with discriminant $d\leq x$ which have such an extreme class number.

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arXiv2014

The distribution of Euler-Kronecker constants of quadratic fields

We investigate the distribution of large positive (and negative) values of the Euler-Kronecker constant $γ_{\mathbb{Q}(\sqrt D)}$ of the quadratic field $\mathbb{Q}(\sqrt{D})$ as $D$ varies over fundamental discriminants $|D|\leq x$. We show that the distribution function of these values is very well approximated by that of an adequate probabilistic random model in a large uniform range. The main tools are an asymptotic formula for the Laplace transform of $γ_{\mathbb{Q}(\sqrt D)}$ together with a careful saddle point analysis.

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